Naslov
The Varchenko determinant for oriented hyperplane arrangements

Autori
Sošić, Milena

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, ostalo, znanstveni

Izvornik
3rd International Conference On Mathematics “An Istanbul Meeting for World Mathematicians“ / - Istanbul, 2019

Skup
3rd International Conference On Mathematics An Istanbul Meeting for World Mathematicians

Mjesto i datum
Istanbul, Turska, 03.07.2019. - 05.07.2019

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
a hyperplane arrangement, an intersection poset, a quantum bilinear form, the Varchenko determinant, oriented arrangements

Sažetak
In this study we will examine a hyperplane arrangement A in a real affine space, based on Varchenko's paper [1]. Therefore, we will explain its main objects such as a domain, an edge, a vertex, an intersection poset, as well as some of its basic properties. We will first define the weight of the hyperplane and the weight of an edge, and then the quantum bilinear form of the arrangement A. Of particular interest is the Varchenko's theorem that provide the formula for determining the determinant of the bilinear form of the arrangement A. Moreover, our study will be presented on oriented real arrangements in a real affine space. Furthermore, we will introduce the orientation of a real arrangement by the unit normal vector ni to a hyperplane Hi. Consequently, the hyperplane Hi divides an affine space into three parts: to the hyperplane itself, to the open half-space containing ni and to the open half-space that does not contain ni. Then to every open half-space is assigned the appropriate weight. Further, the weight of an edge L* of oriented real arrangement is defined as the product of the weights of all open half-spaces whose closures contain the edge L*. Here we have first taken into account that every hyperplane can be obtained as the intersection of the closures of the half-spaces containing it, and then that each domain of oriented arrangement can be taken as a nonempty intersection of corresponding open half-spaces. Then, with respect to the basis of domains, the entries of the quantum bilinear form of a real oriented arrangement will be monomials composed of weights of the corresponding open half-spaces that contain one domain and do not contain other domain of any pair of domains. Given the fact that the entries of the quantum bilinear form of a real arrangement are monomials composed of weights of all the hyperplanes separating any pair of domains, here we have linked a set of variables of weights of hyperplanes with two sets of variables consisting of weights of open half-spaces that are distinguished between crossing a hyperplane in the positive versus the negative direction. [1] A.N. Varchenko, Bilinear form of real configuration of hyperplanes, Adv. Math. 97 (1993), 110-144.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA